Is there evidence that Lucas intentionally introduced similarites between Episodes I and IV?

A Skywalker boy (Luke/Anakin) lives with his relatives in Tatooine, until he encounters a Jedi master (Obi Wan/Qui-Gon) who makes him his apprentice and takes him away from Tatooine1.

The protagonists help a woman of royalty (Princess Leia/Queen Amidala) against a military group (The Empire/The Trade Federation), and in the process the Jedi Master (Obi Wan/Qui-Gon) is killed by a Sith (Darth Vader/Darth Maul). The Skywalker boy pilots a star fighter (with the help of R2-D2) and saves everybody by blowing up a space station (The Death Star/The Droid Control Ship).

The film ends with a ceremony in which the woman of royalty (Princess Leia / Queen Amidala) decorates the heroes of the battle, one of them being the Skywalker boy (Luke/Anakin).

1In one of the sequels, the Skywalker boy (Luke/Anakin) goes to return to Tatooine to help someone he cares about (Han Solo/His mother).

I have read many jokes about Avatar having the same plot that Pocahontas (among others), but I haven't found any about A New Hope and The Phantom Menace.

1 Answer
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TL;DR: Yes, George Lucas has repeatedly said that he intended the prequel trilogy to parallel the original trilogy in many respects, and critics have spoken about it at some length.

It is similar to (though perhaps slightly different from) what is often referred to as the "Star Wars Ring Theory".

George Lucas:

The interesting thing about Star Wars—and I didn’t ever really push this very far, because it’s not really that important—but there’s a lot going on there that most people haven’t come to grips with yet. But when they do, they will find it’s a much more intricately made clock than most people would imagine.
— George Lucas, Vanity Fair, February 2005

And:

[Star Wars] is purposely written like a piece of music, with themes that repeat themselves in different ways, and ideas that reprise from one generation to the next.
- George Lucas, quoted in Scott Chernoff, The Plot Thickens, "Star Wars Insider", July/August 2002.

And:

Instead of destroying the Death Star [like Luke], [Anakin] destroys the ship that controls the robots. It’s like poetry. Every stanza kind of rhymes with the last one.
- George Lucas, The Beginning: Making Episode I Disc 2, The Phantom Menace DVD

And:

It’s very, very clear in the two trilogies that I’m putting the characters in pretty much the same situations sometimes even using the same dialogue so that the father and son go through pretty much the same experience.
- George Lucas, Audio Commentary, The Phantom Menace DVD

Critics:

[Anne Lancashire, professor of Cinema Studies and Drama at the University of Toronto] contends that Lucas began his “carefully designed interrelationships” between the six films by deliberately basing Menace’s narrative on A New Hope’s:

"Anakin Skywalker (eventual father of Luke) is [Menace’s] version of A New Hope’s Luke, going through narratively similar situations and experiences. Anakin, like Luke, is a young boy on the desert planet of Tatooine, from a “broken” family, who is suddenly given the opportunity to embark on an epic quest involving a beautiful, royal young woman in need of his help and a Jedi knight who becomes his mentor. Like Luke, Anakin accepts the opportunity and is flown through space with his mentor to face a test (for Luke, the Death Star rescue of Leia; for Anakin, a literal test before the Jedi Council). Like A New Hope, the film then ends with the boy’s special powers (including his capacity for friendship and love) permitting him to save his friends from annihilation by destroying an enemy battle station. Details of the narrative also correspond from one film to the other: the Jedi mentor’s advice to the protagonist to rely on his feelings, the death of the mentor in a light-saber duel, the association of allies with ancient sacred ruins."

Also, as Lancashire points out, in repeating the narrative pattern of A New Hope, Lucas deliberately repeats that film’s mythological pattern as well:

"Like the plot of A New Hope, that of [Menace] takes us through the three stages of [Joseph] Campbell’s monomyth: the hero’s departure (on his quest), initiation (testing experiences), and return (the emergence from tests to achieve a final victory). This is also both the plot pattern of each of [Star Wars: Episode V—The Empire Strikes Back (1980)] and [Jedi] (with Empire’s “return” phase completed only at the start of Jedi) and, as well, the overarching pattern of the first-made trilogy as a whole (A New Hope as departure. [Empire] as initiation. [Jedi] as return). The integrating viewer can now perceive that Star Wars 1 through 6 will give us the same pattern arching over all six films, in relation to Anakin as hero: with his departure in [Menace], initiation in episodes 2 – 3, and return in 4 – 6 (beginning with his discovery of his son Luke in 4 – 5, and ending with his self-sacrificial death for Luke, and therefore resurrection, at the end of 6)."

Repeating the patterns of plot and myth, Lancashire argues, gives the saga, among other things, “a sense of repeating, increasingly complex cycles of human experience,” within individual lives, from one generation to the next, and “within the overall movement of Anakin’s entire life from boyhood to death.”

In addition: “[The repeated patterns] also allow, through variations, an emotionally and intellectually complicating emphasis upon difference and change. The broad pattern of human life, from youth to maturity to death, remains constant, but individual circumstances within the pattern inevitably differ, creating different possibilities and problems.”

It’s also worth mentioning that the sense of repeating cycles is not only personal, but also political as the films, taken as a whole, reflect the perpetual rise and fall of democracies (the Republic) and dictatorships (the Empire).
- Source

My own observations (and expansion on the above):

A young Skywalker boy from a broken family, who has enormous Force potential and is an unusually talented pilot for his age, lives on Tatooine until he meets one or more Jedi, one of whom is Obi-Wan.

Episode I - the boy is Anakin, the Jedi are Qui-Gon and Obi-Wan

Episode IV - the boy is Luke, the Jedi is Obi-Wan.

This young Skywalker boy goes on a thrilling adventure involving a young female royal who needs his help.

Episode I - the female royal is Padme

Episode IV - the female royal is Leia.

The Skywalker boy goes with the Jedi to face a test.

Episode I - a literal test before the Jedi Council

Episode IV - the test of rescuing Leia from the Death Star.

A lightsaber duel involving Obi-Wan, an older Jedi mentor, and a Sith, happens within eyesight of a young Jedi trainee; the older Jedi mentor is killed, the young Jedi trainee screams "NO!".

Episode I - the older Jedi mentor is Qui-Gon, the young Jedi trainee is Obi-Wan, the Sith is Darth Maul

Episode IV - the older Jedi mentor is Obi-Wan, the young Jedi trainee is Luke, the Sith is Darth Vader.

The movie's climax involves the young Skywalker participating in a space battle to destroy a round battle station which is orbiting a peaceful planet and save his friends.

Episode I - the battle station is the Trade Federation ship, the planet is Naboo

Episode IV - the battle station is the Death Star, the planet is Yavin.

The end of the film is an award ceremony on some stairs in which the Skywalker boy, R2-D2, the young royal woman, and some aliens take part.

In the second installment of the trilogy, the young Skywalker loses his right hand in a lightsaber battle with a Sith involving Anakin.

Episode II - Anakin's hand is chopped off by Dooku

Episode V - Luke's hand is chopped off by Vader (Anakin).

In the third installment of the trilogy, the young Skywalker gets into a lightsaber fight, and it ends with Anakin losing one or more limbs.

Episode III - Obi-Wan pwns Anakin and lops off both legs and an arm

Episode VI - Luke pwns Vader (Anakin) and lops off an arm.

The trilogy focuses on the young Skywalker and his progression as a Jedi, until Anakin has a change of heart and suddenly switches sides, killing a mentor; this happens around the same time as a lightsaber battle involving Palpatine; a change in identity between Anakin and Vader happens.

Prequel trilogy - Anakin becomes a powerful Jedi, but during a lightsaber battle, he turns on Mace Windu, helps Palpatine kill him, and joins the Sith, and he ceases to be Anakin and becomes Vader

Original trilogy - Luke becomes a powerful Jedi, but during a lightsaber battle, Anakin (now known as Vader) switches sides, and kills Palpatine, becoming a Jedi again, and he ceases to be Vader and becomes Anakin once more.

Minor quibble--"Star Wars ring theory" refers to the specific idea that TPM is supposed to match up with ROTJ, AOTC with ESB, and ROTS with ANH. Although Lucas does talk about the prequels rhyming with the original trilogy, I don't think he's said anything to clearly support that order of rhyming over others (including the one the OP asks about, where TPM is most similar to ANH).
– HypnosiflNov 29 '15 at 18:20

@Hypnosifl - See the third quote from Lucas - "Instead of destroying the Death Star [like Luke], [Anakin] destroys the ship that controls the robots. It’s like poetry. Every stanza kind of rhymes with the last one." - George Lucas, The Beginning: Making Episode I Disc 2, The Phantom Menace DVD
– Wad CheberNov 29 '15 at 18:30

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@Hypnosifl - In biblical hermeneutics, that structure is called "Chiastic", which is technically slightly different from a "ring structure", at least in textual criticism of the bible (since I'm studying textual criticism of the bible, that's the only use of chiastic writing I'm familiar with). Either way, the quotes I've used support the interpretation under discussion here, and the site admits that much of their essay is based on Lancashire's analysis, which focuses on the I-IV parallel, at least in the relevant portions I've borrowed.
– Wad CheberNov 29 '15 at 18:55